Abstract
A quantum entity, as discussed in Sect. 2.12, consists of an inseparable pair, namely, its degree of freedom and state vector. One of the most enigmatic features of a quantum entity is that its degree of freedom can never be directly observed. On attempting to observe its degree of freedom, what one ends up observing is the state vector of the degree of freedom, as illustrated in Fig. 3.3. In fact, as discussed in Sect. 7.9, one does not even observe the state vector; every experiment ultimately observes only the effect of the state vector on the projection operators, which are physical detectors. Quantum probability assigns probabilities to the likelihood of a projection operator detecting the state vector.
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Notes
- 1.
It should be noted that there is no experimental proof that von Neumann’s postulate is correct.
- 2.
The more general statement is that the eigenvalues λ n can be reconstructed from the collection of all the counter readings \(x_{i};\ i = 1, 2,\,\ldots,\,N\). The assumption that λ n depends only on x n is made for simplicity.
- 3.
Schrödinger illustrated the paradox of superposing macroscopic states by the famous cat example. A device releases a poison, if triggered by the (uncertain) alpha decay of an unstable atom, that kills a cat. Before the cat is observed, the state of the cat is the following superposed state, namely, \(\vert \text{cat}\rangle = \vert \text{cat; dead}\rangle \vert + \vert \text{cat; alive}\rangle\). The paradox lies in explaining how can the cat be dead and alive at the same time.
- 4.
More advanced books on the foundations of quantum mechanics take issue with this interpretation arguing that a measurement in fact leads to UρM U † that brings back the mixing of all the detector states [4]. The transformation by U can, in principle, be undone by a rotation of the basis states being used to make the measurements.
- 5.
One can create a more complicated experiment where a subsequent measurement is performed on the final state with another device and come to the same conclusion as the thought experiment.
- 6.
Note the notation used implies that \(\mathcal{O}_{E\vert \chi } = E_{\chi }[\mathcal{O}_{E}]\) is an operator on \(\mathcal{V}_{\mathrm{D}}\) and not equal to \(E_{\chi }[\mathcal{O}]\), which is a real number.
- 7.
A partial erasure of the symbol is also possible, as discussed in Sect. 8.8 on the Quantum Erasure.
- 8.
Static electric fields alone cannot act as a trap since the electron will drift along the direction of the electric field and finally hit the electric charge that is the source of the electric field.
- 9.
This asymmetry of time and position is resolved in relativistic quantum field theory by “demoting” position from being a degree of freedom, as is the case for nonrelativistic quantum mechanics, to being a parameter like time; both space and time coordinates label the quantum field’s degrees of freedom. Thus, having both \(t,\vec{x}\) as parameters allows one to have exact relativistic invariance under Lorentz transformations on \(t,\vec{x}\).
- 10.
All three concepts of time, namely, external, intrinsic, and observed time, can be employed to study a quantum process [28].
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Baaquie, B.E. (2013). Quantum Theory of Measurement. In: The Theoretical Foundations of Quantum Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6224-8_9
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